# NEAM 2017: Second Northeastern Analysis Meeting

## October 13-15, 2017, Albany, NY, USA

### Titles and abstracts of participants' talks

10/10/17:

Here is a PDF file with the conference schedule and the titles and abstracts of all participants' talks in alphabetical order: PDF file

### Titles and abstracts of plenary talks

Murat Akman, University of Connecticut: A Minkowski problem for nonlinear capacity

Abstract: The classical Minkowski problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness and regularity.

In this talk, we study a Minkowski problem for certain measure associated with a compact convex set E with nonempty interior and its A-harmonic capacitary function in the complement of E. Here A-harmonic PDE is a non-linear elliptic PDE whose structure is modeled on the p-Laplace equation. If μE denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure μ on Sn-1, find necessary and sufficient conditions for which there exists E as above with μE = μ. We will discuss the existence, uniqueness, and regularity of this problem in this setting.

Lew Coburn, University at Buffalo SUNY: Toeplitz quantization

Abstract: I discuss some recent work with Wolfram Bauer and Raffael Hagger. Here, ℂn is complex n-space and, for z ∈ ℂn, we consider the standard family of Gaussian measures dμt(z) = (4\pi t)-n exp(-|z|^2/4t) dv(z), t > 0 where dv is Lebesgue measure. We consider the Hilbert space Lt2 of all dμt-square integrable complex-valued measurable functions on ℂn and the closed subspace of all square-integrable entire functions, Ht2. For f measurable and h ∈ Ht2 with fh ∈ Lt2, we consider the Toeplitz operators Tf(t) h = P(t)(fh) where P(t) is the orthogonal projection from Lt2 onto Ht2. For f bounded (f ∈ L) and some unbounded f, these are bounded operators with norm ∥‧∥t.

For f, g bounded, with "sufficiently many" bounded derivatives, there are known deformation quantization conditions,

(0) limt → 0 ∥Tf(t)t = ∥f∥

(1) limt → 0 ∥Tf(t) Tg(t) - Tfg(t)t = 0.

We exhibit bounded real-analytic functions f, g so that (1) fails. On the positive side, for the space VMO of functions with vanishing mean oscillation, we show that (1) holds for all f, g in the sup-norm and complex-conjugate closed algebra A = VMO ∩ L and, in fact, A is the largest such subalgebra of L. (1) also holds for all f, g ∈ UC (uniformly continuous functions, bounded or not) while (0) holds for all f ∈ L.

Raúl E. Curto, University of Iowa: Toral and spherical Aluthge transforms

Abstract: We introduce two natural notions of multivariable Aluthge transforms (toral and spherical), and study their basic properties. In the case of 2-variable weighted shifts, we first prove that the toral Aluthge transform does not preserve (joint) hyponormality, in sharp contrast with the 1-variable case. Second, we identify a large class of 2-variable weighted shifts for which hyponormality is preserved under both transforms. Third, we consider whether these Aluthge transforms are norm-continuous. Fourth, we study how the Taylor and Taylor essential spectra of 2-variable weighted shifts behave under the toral and spherical Aluthge transforms; as a special case, we consider the Aluthge transforms of the Drury-Arveson 2-shift. Finally, we discuss the class of spherically quasinormal 2-variable weighted shifts, which are the fixed points for the spherical Aluthge transform.

Francesco Di Plinio, University of Virginia: Maximal averages and singular integrals along vector fields in higher dimension

Abstract: It is a conjecture of Zygmund that the averages of a square integrable function over line segments oriented along a Lipschitz vector field on the plane converge pointwise almost everywhere. This statement is equivalent to the weak L2 boundedness of the directional maximal operator along the vector field. A related conjecture, attributed to Stein, is the weak L2 boundedness of the directional Hilbert transform taken along a Lipschitz vector field. In this talk, we will discuss recent partial progress towards Stein’s conjecture obtained in collaboration with I. Parissis, and separately with S. Guo, C. Thiele and P. Zorin-Kranich. In particular, I will discuss the recently obtained sharp bound for the Hilbert transform along finite order lacunary sets in all dimensions, the singular integral counterpart of the Parcet-Rogers characterization of Lp boundedness for the directional maximal function in higher dimensions.

Ben Dodson, Johns Hopkins: Global well-posedness and scattering for the cubic wave equation in three dimensions

Abstract: In this talk we discuss a global well-posedness and scattering result for the cubic wave equation in three dimensions. We prove this for initial data lying in a space that is critical under the scaling.

Guihua Gong, University of Puerto Rico: On the classification of unital simple separable nuclear C* algebras

Abstract: C* algebras, as noncommutative spaces, have significant applications in the study of differential geometry, topology of manifold and dynamical systems. Simple C* algebras can be regarded as noncomutative single point spaces, which are basic building blocks in the theory of C* algebras. Recent year, there are several important breakthroughs in the Elliott program of classification of simple nuclear C* algebras. In this talk, I will present the complete classification of unital simple separable C*algebras of finite nuclear dimension due to Kirchberg-Phillips, Gong-Lin-Niu, Elliott-Gong-Lin-Niu, and Tikusis-White-Winter.

Stefanie Petermichl, University of Toulouse: On the matrix A2 conjecture

Abstract: The condition on the matrix weight W that is necessary and sufficient for the boundedness of the Hilbert transform acting on vector functions in the matrix weighted L2 space, the matrix A2 condition, is known since 1997 (Treil-Volberg). One motivation for this question stems from the understanding of past and future in multi-variate stationary stochastic processes. Good or sharp quantitative norm control depending on the A2 characteristic of the weight in the 'scalar' case dates to 2007 (P.). Despite notable improvement since then and many new techniques that apply in the scalar setting, the matrix A2 question for the Hilbert transform remains unsolved. We present the best to date estimate (Nazarov-P.-Treil-Volberg), via the use of certain convex bodies and a domination principle in the area. We also present the first sharp estimate of a singular operator in this setting, the matrix weighted square function (Hytonen-P.-Volberg).

Mihai Putinar, UC Santa Barbara: Positivity transformers

Abstract: The question what operations on the distance function of a metric space turn the latter into an isometric subset of a Hilbert space prompted around 1930s a thorough study of distance geometry with positive matrices techniques.

Out of this context Schoenberg, von Neumann and Loewner produced fundamental results. The next generation of harmonic analysts extended the same circle of ideas to homogeneous spaces. The interplay between function theory (Laplace transform methods and interpolation of series of exponentials) with purely matrix analysis tools was prevalent in the area during many decades. Very recently, statisticians found useful these partially forgotten gems of last century analysis, in their attempt to condition and analyze correlation matrices of large systems of random variables.

The talk will be focused on the main historical moments of matrix positivity preservers and will touch some recent progress made possible thanks to symmetric function theory.

Based on joint work with Alex Belton, Dominique Guillot and Apoorva Khare.

Eric Sawyer, McMaster University: A two weight local Tb theorem for the Hilbert transform

Abstract: We prove that the Hilbert transform is bounded from one weighted L2 space to another if and only if the A2 and energy conditions hold, as well as testing conditions taken over weakly accretive families with Lp integrability, p>2. This is joint work with Chun-Yen Shen and Ignacio Uriarte-Tuero.

Wilhelm Schlag, University of Chicago / IAS Princeton: Structure theorems for intertwining wave operators in three dimensions

Abstract: In the 1990s Kenji Yajima carried out a comprehensive analysis of the Lp boundedness of the classical wave operators from scattering theory. In recent joint work with Marius Beceanu, we obtained a representation of the wave operators in R3 as a superposition of translations and reflections. This work combines elements of Yajima's work with methods from harmonic analysis. Specifically, we rely on both Stein-Tomas restriction theory of the Fourier transform, and Wiener's theorem on inversion in a convolution algebra, albeit a complicated one. The latter is essentially a summation method and is used to sum a Born series with large terms. The necessary condition for invertibility needed in Wiener inversion comes from spectral theory developed about 13 years ago. In effect, this amounts to the classical Agmon-Kato-Kuroda theory but completely redone by means of Stein-Tomas type results.

Daniel Tătaru, UC Berkeley: Energy-critical Yang-Mills

Abstract: The hyperbolic Yang-Mills equation is one of the fundamental geometric nonlinear wave equations. The aim of this talk will be to provide an overview of the recent work, joint with Sung-Jin Oh, whose aim is to provide a proof of the Threshold Conjecture for Yang-Mills. This asserts that global well-posedness and scattering holds for all solutions below the ground state energy.

Sergei Treil, Brown: Finite rank perturbations, Clark model, and matrix weights

Abstract: For a unitary operator U all its contractive perturbations U+K, ∥U+K∥≤1 by finite rank operators K with a fixed range R, Ran K ⊂ R can be parametrized by
TΓ := U + B (Γ-Id) B*U, Γ:ℂd → ℂd, ∥Γ∥≤ 1,
where B is a fixed unitary operator B: ℂd → R.
Under the natural assumptions that R is a *-cyclic subspace for U and Γ is a strict contraction, the operator TΓ is the so-called completely non-unitary (c.n.u) contraction, so it is unitarily equivalent to its Sz.-Nagy--Foiaș functional model.
The Clark operator is a unitary operator that intertwines the operator TΓ (which we assume is given to us in the spectral representation of the operator U) and its functional model. Description of such operator is the subject of Clark theory.
I will completely describe the Clark operator in its adjoint for the general finite rank perturbations. The adjoint Clark operator is given by the vector-valued Cauchy transform, and the direct Clark operator is given by simple algebraic formulas involving boundary values of functions from the model space. Weighted estimates with matrix valued weights appear naturally in this context.
The case of rank one perturbation of a unitary operator with purely singular spectrum was completely described (from a different point of view) by D. Clark, and later further developed by A.Aleksandrov and then by A. Poltoratski; the case of general rank one perturbations was resolved by Liaw-Treil.
In the case of perturbations of rank d>1, some new phenomena requiring careful investigation appear, and are resolved.
The talk is based on a joint work with C. Liaw.

Monica Vișan, UCLA: Almost sure scattering for the cubic NLS in four dimensions

Abstract: I will discuss recent work with R. Killip and J. Murphy on almost sure scattering for the energy-critical NLS in four space dimensions with radial randomized initial data.